Mathematical Formalisms: Categorical Foundations for Matter–Information Correspondence

We formulate the categorical grammar underlying a modular four-law programme for emergent spacetime dynamics. Law I, presented here, is the foundational layer: it fixes the language in which Laws II (phase-bound matter), III (frequency-modulated processes), and IV (information-bearing structures) are written. We argue that the appropriate mathematical setting for a matter–information correspondence is that of symmetric monoidal (∞,n)-categories together with their sheaf-theoretic, operadic, and type-theoretic refinements. We isolate eight composition hooks—formal interfaces marked H1–H8—that downstream laws plug into without modification. We define the matter–information functor M : Phys → Info as a strong monoidal, dagger-preserving functor between symmetric monoidal dagger categories, and we prove three structural results.